How Do You Visualize the Number Line?
 

Just a note about an interesting article in the May 30 issue of the journal [I]Science[/I]. Below is the layman's abstract from "This Week in Science"; can't link to the actual article because it requires a subscription.

[QUOTE]Concepts of number and of space seem so fundamental to human experience as to be deeply embedded in cognitive structures. In general, young, preschool children intuitively map quantities onto a line where, in most cases, small numbers are placed to the left and large numbers to the right. The scale used in this mapping is logarithmic and not linear, meaning that 10 is placed in the middle of a line spanning 1 to 100. With the advent of schooling, along with exposure to cultural instantiations of number words such as rulers, the mapping shifts to a linear representation. From the latest in a series of visits to an Amazonian indigenous people, the Mundurucu, who lack formal number systems, Dehaene et. al. (p. 1217 of 30-May-2008 issue) have obtained evidence that in these people the logarithmic mapping seen in youngsters persists in adults. Thus the log-to-linear shift depends on culture- the existence of integer words separated by unit increments (for instance, twenty-three and twenty-four) and/or education in linear arithmetic operations such as addition and subtraction. [/QUOTE]

What I find fascinating is that for pre-schoolers, the number line is approximately logarithmic, not linear. It's surprising at first, but in hindsight, that makes perfect sense.

Norm

Negatives on my left, positives on my right.

When I'm thinking about a relatively short interval, it's linear, but when I'm "pulling back" for a broader view, higher-magnitude numbers slide in from right and left as the scale adjusts logarithmically.

To take a photo of the entire (straight) number line
I would use a fish-eye lens.

For numbers <100 I usually visualize them like this:

20 21 22 23 23 ...
19
18
17
16
....
2
1
0
-1
-2
-3
...

Maybe I saw a chart up to 20 when I was small, then another when I was older and combined the two?
5 is linked to 25 by a blurred diagonal line, 6 is linked to 36 etc.


Years A.D. look like this:

1900 1901 1902 1903 ... 2000
1800
1700
...

i.e. Before 1900 is much more compressed. There is a wall just after 2000 and the numbers then spread out above and below the number line.

Any recommended medication? :help:

[QUOTE=Flatlander;140328]Any recommended medication? :help:[/QUOTE]

None. That's the way they do that in flatland, I presume. H.

[QUOTE=Spherical Cow;136551]Just a note about an interesting article in the May 30 issue of the journal [I]Science[/I]. Below is the layman's abstract from "This Week in Science"; can't link to the actual article because it requires a subscription.



What I find fascinating is that for pre-schoolers, the number line is approximately logarithmic, not linear. It's surprising at first, but in hindsight, that makes perfect sense.

Norm[/QUOTE]

Well, yes. The reason is obvious: positional notation.
100 is 10 times bigger than 10, but as written, is only 1 additional
character. People wind up ordering numbers not by magnitude, but
lexicographically.

[QUOTE=R.D. Silverman;140646]Well, yes. The reason is obvious: positional notation.
100 is 10 times bigger than 10, but as written, is only 1 additional
character. People wind up ordering numbers not by magnitude, but
lexicographically.[/QUOTE]How many pre-schoolers can read?

Paul

[QUOTE=xilman;140652]How many pre-schoolers can read?

Paul[/QUOTE]

I suspect that most of them have seen the numbers 1, 10, 100, etc.

[QUOTE=R.D. Silverman;140671]I suspect that most of them have seen the numbers 1, 10, 100, etc.[/QUOTE]Perhaps. I'm not so sure that they comprehended the symbolism.

However, they've probably heard words of about the same length being used to describe quantities at least 10 times bigger: hundred, thousand, million, etc.; so that may be related.


Paul

I don't visualise a number line at all, why would I need to?

I don't visualise the alphabet in a line when I'm writing either.

My guess would be that it has something to do with our inate abilities of scale/distance. For instance, back in our hunter/gather days, if you were hunting an animal for example, knowing the animal was 10 or 11 feet away wouldn't really make a difference, but if it was 1 ft or 10 ft then it would make a huge difference, same as between 10 and 100. Not that our brains knowthe numbers naturally, but our minds do know relative distances.

This is all a guess. I tried it on my 5 year old and he placed the 10 in the center of the line, just like the article said.


Matt